# Questions tagged [hamiltonian]

The central term in the hamiltonian formalism. Can be interpreted as an energy input, or "true" energy.

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### PDE from Hamiltonian density

For the wave equation Hamiltonian density is $2H=\phi_t^2+\phi_x^2$ while the Lagrangian density is $2L=\phi_t^2-\phi_x^2$. I can easily compute the pde from the Lagrangian density but how does one do ...
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### Derivation of $U=e^{-iHt/\hbar}$ [on hold]

When trying to derive the time evolution operator, I arrive to this differential equation: $$\frac{\partial}{\partial t}{\hat U(t,t_{o}})=\frac{-i}{\frac{h}{2\pi}}{\hat{H}}{\hat U(t,t_{o}})$$ How ...
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### Dyson Series Iteration - Gives Exact Solution?

When we derive the Dyson series for usage as the time evolution operator in the case of a time dependent Hamiltonian, we start with the equation: \begin{align}\hat{U}_I(t,t_i) = 1 - \frac{i}{\hbar}\...
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### Why is the generalized momentum replaced by the momentum operator but not the ordinary momentum?

I was trying to understand the derivation of the Hamiltonian for a charged particle in an electromagnetic field. https://en.wikipedia.org/wiki/Hamiltonian_mechanics#...
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### Significance of energy in a time dependent quantum box

The Hamiltonian for a particle in a finite box is $$H = \frac{p^2}{2m} + V(x)$$ which will give time evolution as $$i\hbar d/dt|{\psi(t)}\rangle = H|{\psi(t)}\rangle \, .$$ However, if I do a ...
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### Direct Derivation of Kraus Operator from Interaction Hamiltonian

For the dynamics of open quantum systems, the Kraus operators $K_\kappa$ can be derived from the unitary orbit $U(t)\rho U(t)^\dagger$ for $\rho=\rho_S\otimes\rho_E$ of the composite system given by ...
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### Completeness condition involving continuum states

Consider a potential $V(x)$ in 1d. Suppose that $V(|x| > a )= 0$ for some positive $a$. We then know that the hamiltonian $H = - \frac{\partial^2}{\partial x^2 } + V(x)$ has non-normalizable or ...
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### (Altland-Simon) Deriving ferromagnetic interaction term from interacting tight-binding Hamiltonian

Below is a part of the book "Condensed Matter Field Theory" by Altland and Simon. My question is about deriving the equation with red arrow. This is outlined in the exercise in the figure, but I don'...
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### Prove conservation law in quantum mechanics

I major in Math, and I am studying Quantum Mechanics (QM). I see the conservation law in QM as a mathematical theorem. Please check if my understanding is right, and help me to prove the theorem? ...
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### Manipulating Dirac Notation

I have trouble getting my head around manipulating Dirac notation, it's still new to me and I'm not used to it. I'm following the rotating wave approximation derivation for Rabi oscillations and light ...
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### Calculating exact energy levels of perturbed Hamiltonian

I wish to find the exact energy levels of the following perturbed hamiltonian. $$\hat{H}=\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2+\alpha x+\beta p^2.$$ I believe that it can be solved by using the ...
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### Simultaneous eigenstates of Hamiltonian and momentum operator

Given the potential barrier, \begin{align} V(x, y) = \left\{ \begin{array}{cc} V_{0} & \hspace{5mm} \text{if $0 \leq x \leq D$} \\ 0 & \hspace{5mm} \text{...
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I have some problems understanding the Lagrangian and the Hamiltonian formalism. Those can be condensed in the following "derivation" of $\frac{\partial T}{\partial t} = 0$ from the equation $\frac{\... 1answer 71 views ### Is there always a canonical transformation such that the new Hamiltonian only depends on the new momenta? Given the Hamiltonian$H(x,p)$of a system. Is there always a coordinate transformation such that the new Hamiltonian is$K(x',p')=K(p')$? 2answers 32 views ### Hamiltonian and Supercharges Mirror Symmetry p.188 Eq. 10.109 states that $$H \left\vert \alpha\right> = 0 \Longleftrightarrow Q \left\vert\alpha\right> = \overline{Q} \left\vert\alpha\right> =0. \tag{10.109}$$ I dont ... 0answers 21 views ### Evolution of the propagator in the Interaction picture? The evolution operator in the interaction picture is defined as$U_I=e^{iH_0t}e^{-iH_St}e^{-iH_0t}$Where$H_S=H_0+V$I am trying to find the evolution of the operator$U_I$. In literature it is ... 1answer 71 views ### Can we craft a Hamiltonian such that the measurement is consistent with the discrete measurement taught in Quantum physics? So the way I understand this, the way measurement is taught is that you have a wave function$\Psi(t)$. It's evolution over time is : $$i \hbar \frac{d}{d t}\vert\Psi(t)\rangle = \hat H(t)\vert\Psi(t)... 1answer 210 views ### Couple of non-interacting, non-integrable Hamiltonian systems I have two Hamiltonians, H(p_1, q_1, \dots, p_n, q_n) and H'(p_{n+1}, q_{n+1}, \dots, p_m, q_m). They are non-interacting, indeed they have different variables. Moreover, I know that they are both ... 2answers 10k views ### Commutation of Hamiltonian with momentum In which case does the Hamiltonian H commutes with the momentum P? Can anybody help me? With an example? (No particular or strange Hamiltonians and no particular momenta are involved). How can I ... 1answer 86 views ### Quantum field theory , Schrödinger wavefunction \psi is a state that given |\psi\rangle=\int d^3x\psi(x)|x\rangle How does the wave function change in time? The Hamiltonian can be written as$$H=\int d^3x\frac{1}{2m}\nabla\psi^*\nabla\psi=\int ... 0answers 31 views ### Generically, why do we want to evolve states with unitary operators? [duplicate] Why is it so important that operators that evolve states are unitary? 0answers 13 views ### How to know the symmetries of a coupling in the Hamiltonian Suppose you have an interaction term in your hamiltonian that looks like \begin{equation} H=\sum_{ijkl}U_{ijkl}c^\dagger_ic^\dagger_jc_kc_l \end{equation} where$U$is the coupling and$c$,$c^\...
When we calculate the band structure of some solid then we often find that in the bottom of the conduction band the dispersion looks approximately quadratic with some new effective mass: E(k) = \...
Say we had a set of wave functions $\psi(x,t)$ that we new the values of for all $x$ and $t=t_0..t_1$. Say we had $N$ of these wavefunctions, perhaps $N=10$. All these wave functions start off at ...